# Difference between revisions of "Math 22 Continuity"

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Then, <math>\lim_{x\to 3^+} f(x)= \lim_{x\to 3^+} 14-x^2=14-(3)^2=5</math>. | Then, <math>\lim_{x\to 3^+} f(x)= \lim_{x\to 3^+} 14-x^2=14-(3)^2=5</math>. | ||

− | Since <math>\lim_{x\to 3^-} f(x)= 5 = \lim_{x\to 3^+} f(x)</math>, \lim_{x\to 3 | + | Since <math>\lim_{x\to 3^-} f(x)= 5 = \lim_{x\to 3^+} f(x)</math>, \lim_{x\to 3} f(x) exists. |

− | Also notice <math>f(3)=14-(3)^2=5=\lim_{x\to 3 | + | Also notice <math>f(3)=14-(3)^2=5=\lim_{x\to 3} f(x)</math> |

So by definition of continuity, <math>f(x)</math> is continuous at <math>x=3</math>. | So by definition of continuity, <math>f(x)</math> is continuous at <math>x=3</math>. |

## Revision as of 08:20, 16 July 2020

## Continuity

Informally, a function is continuous at means that there is no interruption in the graph of at .

## Definition of Continuity

Let be a real number in the interval , and let be a function whose domain contains the interval . The function is continuous at when these conditions are true. 1. is defined. 2. exists. 3. If is continuous at every point in the interval , then is continuous on theopen interval.

## Continuity of piece-wise functions

Discuss the continuity of

On the interval , and it is a polynomial function so it is continuous on

On the interval , and it is a polynomial function so it is continuous on

Finally we need to check if is continuous at .

So, consider

Then, .

Since , \lim_{x\to 3} f(x) exists.

Also notice

So by definition of continuity, is continuous at .

Hence, is continuous on

## Notes

*Polynomial function* is continuous on the entire real number line (ex: is continuous on )

*Rational functions* is continuous at every number in its domain. (ex: is continuous on since the denominator cannot equal to zero)

**This page were made by Tri Phan**